Wonderful Mobius Transformation Video

Jun 25 2007 Published by under topology

Via The Art of Problem-Solving, a great video on Mobius transformations. I never really got how the inversion transformation fit in with the others before seeing this!

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  • Xanthir, FCD says:

    That's awesome, Mark! It's really interesting how geometry all fits together.
    I wonder if the sphere analogy was intended originally, or if it was merely realized later that the Mobius transformations correspond to simple movements on a sphere?

  • revere says:

    Very, very cool. This is the first time I "got it."

  • Canuckistani says:

    Wow. Just wow. Cam we have the math behind the video?

  • Krantz, S. G. "Möbius Transformations." §6.2.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 81, 1999.
    Needham, T. "Möbius Transformations and Inversion." Ch. 3 in Visual Complex Analysis. New York: Clarendon Press, pp. 122-188, 2000.
    Weisstein, Eric W. "Möbius Transformation." From MathWorld--A Wolfram Web Resource.

  • Superb use of visuals. As it was for you, this generated an "aha!" moment for me. I love "aha!" moments.

  • Josiah Carlson says:

    That is a great find.

  • Crow says:

    Wonderful. This is one of the best pieces of mathematical educational material I've ever seen. But more than that, this is one of those "isn't the universe we live in cool" moments for me. When you can combine education with that kind of wonder and delight, you've really got something good.
    Huge kudos to the authors.

  • Vlad Shcherbina says:

    Overwhelming! I feel burning interest to complex analysis after your movie. A great moment of enlightment (both literally and allegorically). Thanks.

  • CCP says:

    I don't know from math, but this was really beautiful on a purely visual level. Too bad about the sedate piano accompaniment...those colors seem more suited to an Anthem of the Sun-era Grateful Dead soundtrack, man.

  • Agnostic says:

    There's a YouTube account for visuals used in a topology seminar: http://www.youtube.com/user/bothmer
    They're pretty cool too. Since it's for a seminar, they also provide a more rigorous explanation in the description box.

  • Stephen Wells says:

    Two minutes ago I didn't even know what a Mobius transformation was. Seeing that inversion in the plane corresponds to rotating the projective sphere- that was truly enlightening. Does this generalise to higher dimensions?- e.g. can I think of an inversion in 3-D space as corresponding to rotation of a projective hypersphere?

  • Very nice indeed. Some years ago, I tried to do similar things for the Hyperbolic Plane, but while I came up short, these guys succeeded brilliantly. The Schumann was good accompaniment, but I too would have preferred something a little jumpier. Maybe Feltsman doing a fugue from the WTC.

  • That is the type of music that all maths should be set to, ever. From now on, I'm not learning anything without a piano accompaniment.

  • Doug says:

    I second the Jonathan Vos Post #4 reference to Tristan Needham, 'Visual Complex Analysis' and his use of "amplitwisting".
    There is a website with PDF extracts:

  • Paul Carpenter:
    I dunno — I think this video, in particular, would have gone very well with a Kraftwerk track like "Neon Lights".
    By the way, those who want more details should check out the Secret Blogging Seminar's comments on this video.

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